Numerical Basis of Calculating the Potential

Based on lecture given by Dilano K. Saldin, Univ. of WI-Milwaukee
October - November 1998

The starting point is the set of free atom radial wave functions Rl(r)
tabulated by Herman-Skillman. For a closed subshell (electron occupancy
= 2l+1), the electron density can be written - using spherical symmetry

.

This is normalized by the condition (extra factor of 2 due to electron
spin)

where Nnl is the number of electrons in the subshell specified
by the quantum numbers n and l.

Note that even in the case of the outer unfilled shells we have to perform
the same sum over the magnetic quantum number since the occupation of each
of the magnetic quantum number states must be regarded as equally likely.
With the above normalization, the total electron charge density due to
each atom is

.

The Coulombic contribution to the resulting potential of the atom, in
Hartree units, is

where the first term is the contribution from the nucleus of charge
Z and the second term is the electronic contribution (u(r) < 0) defined
by

.

Since r(r) and u(r) depend only on the radial
coordinate r, we need only the radial part of the Laplacian. This allows
us to redefine

.

Thus, Poisson's equation becomes

.

For a given r(r), this equation can be solved
for u(r) (for details on this, see Appendix 1 of Loucks' paper). Hence,
the total Coulombic contribution to the atomic potential V(r) can be found.